Nombres réels de complexité sous-linéaire : mesures d'irrationalité et de transcendance

Abstract

By means of a quantitative version of the Schmidt Subspace Theorem, we obtain irrationality and trancendence measures for real numbers whose expansion in an integer base has a sublinear complexity. We further give several applications of our general results to Sturmian, automatic, and morphic numbers, and to lacunary series. In particular, we extend a theorem on Sturmian numbers established by Bundschuh in 1980. We also provide a first step towards a conjecture of Becker by proving that irrational automatic real numbers are either S- or T-numbers. This improves upon a recent result of Adamczewski and Cassaigne, who established that irrational automatic real numbers cannot be Liouville numbers.